the persistence and rigidity of wages and prices
TRANSCRIPT
The persistence and rigidity of wages andprices∗
Boris Hofmann† and Matthias Paustian‡
November 1, 2004
Abstract
We analyze welfare effects of monetary policy rules when both pricesand wages are sticky using the linear-quadratic framework of Rotemberg andWoodford (1997). .... Among some simple rules that perform reasonablywell across structural assumptions is a first difference version of the classicalrule proposed by Henderson and McKibbin (1993) and Taylor (1993).
1 Introduction
The Philips Curve has been a central piece in macroeconomics for decades. Theoriginal work by Philips related wage inflation to unemployment. Starting frommicrofounded models with sluggish price adjustment, modern macroeconomicshas derived the so called New Keynesian Philips curve relating price inflation tomarginal cost and expected future price inflation. This relationship is at the cen-ter of large strand of literature, studying welfare based optimal monetary policysummarized in Woodford (2003, Chapter 6).
Since the seminal paper by Erceg, Henderson, and Levin (2000), it is wellknown that the joint analysis of sticky prices and wages has important implica-tions for the conduct of monetary policy. In particular, a trade off between thestabilization of wage inflation, price inflation and the output gap exists for anykind of shock hitting the economy. Recently, a number of important contributions
∗This research was started while the second author was visiting the Research Department ofthe Deutsche Bundesbank. We would like to thank the Deutsche Bundesbank, especially HeinzHerrmann, for kind hospitality. All errors are our own
†Deutsche Bundesbank, Email:[email protected]‡Center for European Integration Studies (ZEI), Walter-Flex-Str. 3, 53113 Bonn, Germany and
Bonn Graduate School of Economics Email:[email protected]
1
have pointed to wage stickiness as a crucial feature that allows monetary generalequilibrium models to match the data. In this sense, empirical studies emphasiz-ing wage rigidities, like Christiano, Eichenbaum, and Evans (2004) or Smets andWouters (2003) render the joint analysis of sticky wages and sticky prices highlyrelevant.
Fuhrer and Moore (1995), Fuhrer (1997) and a number of other authors havecriticized the New Keynesian Philips curve for its inability to capture the persis-tence in price inflation and for putting too much emphasis on forward lookingbehavior. In response Galı and Gertler (1999) have included a fraction of rule ofthumb price setters into the standard New Keynesian setup, that gives rise to theso-called hybrid Philips curve. The hybrid Philips curve relates current inflationto last periods inflation, marginal cost and future inflation allowing for much morepersistence in inflation.
An issue largely neglected until now is the joint analysis of hybrid wage andprice Philips curves for optimal monetary policy.1 What role does rule of thumbbehavior in both wage and price setting play for optimal monetary policy? Howis the performance of simple monetary policy rules affected by the fraction ofbackward looking agents in price and wage setting. We take up the question ofoptimal monetary policy with sticky wages and prices posed by Erceg, Henderson,and Levin (2000) and allow for backward looking rule of thumb behavior in bothprice and wage setting. Our hybrid price and wage Philips curves are estimatedon Euro area and U.S. data via GMM. The estimates of the structural parametersare then used in a number of policy questions.2
We derive a purely quadratic welfare based loss function from the model andcompute fully optimal monetary policy under commitment. We show analyticallythat a key parameter governing the relative weight on wage inflation variabilityversus price inflation variability is the Frish elasticity of labor supply. For a rangeof plausible values taken from micro-econometric estimates between 0.25 and 2,the weight on the variance of wage inflation relative to the weight on the vari-ance of price inflation extends from 15 down to 1.5. Given that this parameter iscrucial for optimal monetary policy, it is very unfortunate that it is typically veryimprecisely estimated in macro models as well as in micro studies.
The paper proceeds as follows. Section 2 describes the setup of the modelwhich is similar to Erceg, Henderson, and Levin (2000). Section 3 derives thehybrid wage and price Philips curves from a measure of backward looking wageand price setters and summarizes the key equations determining general equilib-
1Steinsson (2003) as well as Amato and Laubach (2003b) analyze optimal monetary policywith fully flexible wages and a hybrid price Philips curve.
2Amato and Laubach (2003a) have estimated parameters governing the wage and price Philipscurves in a fully specified general equilibrium model, but did not allow for rule of thumb wageand price setters.
2
rium in the model. In section 4 we present our baseline calibration and discussthe crucial parameters that affect the loss function. Section 5 analyzes the welfareeffects of fully optimal monetary policy as well as of certain popular simple rulesfor varying degrees of backward and forward looking wage and price setters. Sec-tion 7 summarizes the findings and concludes. Derivations of the proposition aredeferred to the appendix.
2 Model
The model we consider is very similar in its key building blocks to the one inErceg, Henderson, and Levin (2000), except for rule of thumb wage and pricesetters. In particular, capital is in fixed supply in the aggregate. We assume thatthere exists an economy wide rental market that allows capital to freely movebetween firms. We abstract from aggregate capital accumulation, because thederivation of a welfare based loss function for the central bank becomes extremelycumbersome with aggregate capital accumulation. Edge (2003) shows that theloss function in such a case additionally involves the variance of the investmentgap, the covariance of the investment gap with the output gap and all future auto-covariances of the investment gap. The assumption of perfect mobility of capitalis not innocuous, either. It implies that real marginal cost is the same for allfirms and independent of the quantity produced by any single firm. We relax thisassumption in our robustness check. Finally, we assume that subsidies exists thatcompletely offset the effects of monopolistic competition in the steady state. Theassumption that the economy is efficient in the steady state is again chosen in orderto avoid highly cumbersome derivations of the loss function that would arise withan inefficient steady state.3 In the next subsection we start with a discussion of thehouseholds problem.
2.1 Households
There is a continuum of households with unit mass indexed byh. Households areinfinitely lived, supply laborNt(h) and receive nominal wageWt(h), consumefinal goodsCt(h), purchase state contigent securitiesBt(h). Furthermore, they aresubject to lump sum transfersTt, hold nominal money balancesMt(h) and receiveprofitsΓ(h)t from the monopolistic retailers. The utility function is assumed tobe separable in consumption, real money balances and leisure. The representative
3 This highly demanding task has recently accomplished by Benigno and Woodford (2004).
3
household’s problems is:
maxBt+1,Mt+1,Ct,Nt
Et
∞∑i=0
βt+i
[U(Ct+i(h)) +H
(Mt+i(h)
Pt+i
)+ V (Nt+i(h))
]s.t. Ct(h) =
δt+1,tBt(h)−Bt−1(h)
Pt
+ (1 + τw)Wt(h)
Pt
Nt(h) + Tt(h) + Γt(h)
− Mt+1(h)−Mt(h)
Pt
.
HereBt is a row vector of state contingent bonds, where each bond pays oneunit in a particular state of nature in the subsequent period. The column vectorδt+1,t represents the price of these bonds. Therefore, the inner product gives totalexpenditures for state contingent bonds.Bt−1 is number of state contingent bondsthat pay off in the particular state of nature at timet. The first order conditions forconsumption and state contingent bond holdings give rise to the standard Eulerequation. Note that consumption is perfectly insured against idiosyncratic laborincome and therefore consumption is no longer indexed byh.
UC(Ct) =Etβ
{Pt
Pt+1
Uc(Ct+1)
}Rn
t . (1)
We follow the standard practice to omit the first-order condition for money hold-ings as this equation merely serves to back out the quantity of money that supportsa given nominal interest rate.
The following functional forms are used in later parts of the analysis
U(Ct) =C1−σ
t
1− σ(2)
V (Nt) = − N1+χt
1 + χ(3)
A continuum of households supply differentiated laborNt(h), which is aggre-gated according to the Dixit-Stiglitz form:
Lt =
[∫ 1
0
[Nt(h)]κ−1
κ dh
] κκ−1
. (4)
HereWt is the standard Dixit-Stiglitz index. The demand function for differenti-ated labor is:
Nt(h) =
[Wt(h)
Wt
]−κ
Lt (5)
4
With probability θw a randomly chosen household is allowed to set its nominalwage in a given period. The household maximizes expected utility through choiceof the nominal wage subject to the demand curve and the budget constraint. TheFOC for this problem is:
Et
∞∑j=0
(θwβ)jNt+j(h)UC(Ct+j)
[(1 + τw)
W ∗t (h)
Pt+j
+κ
κ− 1
VN(Nt+j(h))
UC(Ct+j)
]= 0
(6)
2.2 Production
Firms in the final good sector produce a homogeneous good,Yt, using intermedi-ate goods,Yt(z). There are a continuum of intermediate goods a measure unity.The production functions that transforms intermediate goods into final output isgiven by
Yt =
[∫ 1
0
Yt(z)ε−1
ε dz
] εε−1
(7)
whereε > 1. The solution to the problem of optimal factor demand yields thefollowing constant price elasticity demand function for varietyz.
Yt(z) =
(Pt(z)
Pt
)−ε
Yt (8)
A continuum of monopolistically competitive intermediate goods firms ownedby consumers indexed byz ∈ [0, 1] uses both laborLt(z) and capitalKt(z) toproduce output according to the following constant returns technology:
Yt(z) = AtLt(z)1−αKt(z)
α (9)
whereAt is a technology parameter. Capital is freely mobile across firms ratherthan being firm specific. Firms rent capital from households in a competitivemarket on a period by period basis after they observe the productivity shock. Firmz choosesLt(z) andKt(z) to minimize total cost subject to meeting demand
WtPtLt(z) + ZtKt(z) s.t. AtLt(z)1−αKt(z)
α − Yt = 0. (10)
Let Xt denote the Lagrange multiplier with respect to the constraint andwrt the
real wage. The first order conditions with respect toLt(z) andKt(z) are given by
wrt = (1− α)XtAtK(z)α
t L(z)−αt (11)
Zt =αXtAtK(z)α−1t L(z)1−α
t (12)
5
The first order conditions imply that inputs adjust to equalize marginal cost acrossdifferent factors, where the marginal cost of a factor is the ratio of the factor priceto the marginal product. Since all firms choose the same capital to labor ratio,marginal cost is equalized across firms. This will not be true for the case of firmspecific capital, where the immobility of capital across firms prevents firms fromchoosing equal capital to labor ratios.
For price setting, we follow the widely used time dependent pricing approachof Calvo (1983). In any give period, there is constant probabilityθ of receiving asignal that allows the firm to reset its price. Is the random signal not received, thefirm carries on the price posted in the last period and satisfies any demand at thatprice.4
The problem of a firm that receives a signal to change its price in periodt isto maximize expected real profits as valued by the household in those states of theworld where the price remains fixed.5
maxP ∗t (z)
Et
∞∑i=0
(θβ)iΛt+i
{(1 + τp)
[P ∗t (z)
Pt+i
]1−ε
Yt+i −Xt+i
[P ∗t (z)
Pt+i
]−ε
Yt+i
}(13)
Here,Λt is the households marginal utility of consumptioni periods from nowandτt is sales subsidy suitably chosen as to offset the steady state effects of mo-nopolistic competition(1 + τp = ε
ε−1). The first order condition is
Et
∞∑i=0
(θβ)iΛt+i
{(1 + τp)P
∗t (z)(1− ε) [Pt+i]
ε−1 Yt+i + εXt+iPεt+iYt+i
}(14)
2.3 The flexible price solution and the gaps
In order to obtain welfare in terms of an output gap, it is useful to consider the so-lution under perfectly flexible prices. As shown in proposition 1 in the appendix,up to a first order approximation flexible price output is given by
Y ∗t =
[1 + ω2
ω2 + α− (1− α)ω1
]At (15)
Hereω1 ≡ UCC CUC
is the elasticity of the marginal utility of consumption evaluated
at the steady state andω2 ≡ VNN NVN
is the elasticity of the marginal utility of labor.
4A markup of price over marginal cost is necessary to ensure that for small positive demandshocks, the firm still makes positive profits on the marginal units demanded despite increasingmarginal cost.
5Here we have made use of properties of the Cobb-Douglas Production function, rewritingtotal cost as marginal cost times production.
6
The subutility functionsU(Ct) ≡ C1−σt
1−σimpliesω1 = −σ and forV (Nt) ≡ −N1+χ
t
1+χ
we haveω2 = χ. Given these functional forms, the natural level of output in log-deviation is given by
Y ∗t =
[1 + χ
χ+ α+ (1− α)σ
]At (16)
One can use this equation together with the firm’s first-order condition for labordemand, to derive a key equation for this model. That equation links marginal costto the output gap and the gap between the average marginal rate of substitutionbetween consumption and labor and the real wage.
Xt =
[χ+ α
1− α+ σ
](Yt − Y ∗t
)−[χLt + σYt − wr
t
](17)
When there are no nominal rigidities in the labor market, the marginal rate ofsubstitution between consumption and labor is equal to the real wage and the lastterm in brackets vanishes. We then recover the condition from sticky price modelsthat marginal cost is log-linearly related to the output gap. When additionally,prices are perfectly flexible, the marginal product of labor is equal to the real wage,i.e. log marginal cost is zero and it follows that the output gap is zero. These twogaps, the difference between the real wage and the marginal product of labor andthe difference between the real wage and the marginal rate of substitution are atthe center of the welfare analysis.6
3 Hybrid wage and price Philips curves
We allow for backward looking elements in the wage and price setting as pro-posed by Galı and Gertler (1999) and Galı, Gertler, and Lopez-Salido (2001). Inparticular, we assume that price setters that do receive a signal to re-set their pricebelong to one of two groups. A measureω of backward looking firms set theirprice according to the following rule of thumb
P bt =πt−1
(P ∗t−1
)1−ω (P b
t−1
)ω(18)
The rule posits that these firms adjust prices according a geometric average ofprices changed last period adjusted for last periods inflation rate. The consumptionbased price index is given by
Pt ≡[∫ 1
0
Pt(z)1−εdz
] 11−ε
(19)
6See Gali, Gertler, and Lopez-Salido (2003) for estimating the cost of business cycle variationson the basis of these gaps.
7
Since the fraction of firms that can change the price is chosen randomly and bythe law of large numbers, the aggregate price index evolves as
P 1−εt = θP 1−ε
t−1 + (1− θ)(1− ω)(P ∗t )1−ε + (1− θ)ω(P bt )1−ε (20)
As shown in proposition 3 in the appendix, this setup gives rise to the followinghybrid new Keynesian Philips curve
πt =(1− ω)(1− θ)(1− βθ)
ζXt +
βθ
ζEtπt+1 +
ω
ζπt−1 (21)
Here ζ ≡ θ + ω[1 − θ(1 − β)]. If we assume that capital is no longer freelymobile, but firm specific, proposition 4 in the appendix shows that the PhilipsCurve is given by
πt =(1− ω)(1− θ)(1− βθ)
ζ
(1− α)
(1− α+ αε)Xa
t +βθ
ζEtπt+1 +
ω
ζπt−1 (22)
Here, Xat is average marginal cost. Since capital is fixed at the firm level, the
capital-labor ratio differs across firms and so does marginal cost. For the optimalmonetary policy analysis, we will consider both common and firm specific capital.
Similarly for wage setting, we assume that those households that do receive asignal to re-set their wages belong to one of two groups. A measureϕ of backwardlooking households set their wage according to the following rule of thumb
W bt =πw
t−1
(W ∗
t−1
)1−ϕ (W b
t−1
)ϕ(23)
The wage index is defined as
Wt ≡[∫ 1
0
Wt(h)1−κdh
] 11−κ
(24)
Since the fraction of wage setters that receive the signal to change their wageis randomly chosen and by the law of large numbers, the aggregate wage indexevolves according to the formula
W 1−κt = θwW
1−κt−1 + (1− θw)(1− ϕ) (W ∗
t )1−κ + (1− θw)ϕ(W b
t
)1−κ(25)
As shown in proposition 2 in the appendix This setup gives rise to a hybrid newKeynesian wage Philips curve
πwt =
(1− ϕ)(1− θw)(1− βθw)
(1 + κχ)ζwµt +
βθw
ζwEtπw
t+1 +ϕ
ζwπw
t−1 (26)
Hereζw ≡ θw + ϕ[1− θw(1− β)] andµt ≡ χLt + σCt − wrt.
8
3.1 The key equations
The model has 6 endogenous variables: price inflationπt, wage inflationπwt , la-
bor Lt, outputYt, marginal costXt, and the real wagewrt . We treat the rate of
price inflation as the central bank’s instrument. The policymakers problem is tomaximize the welfare measure through choice of the inflation rate subject to thefollowing constraints.7
πwt =
(1− ϕ)(1− θw)(1− βθw)
(1 + κχ)ζwµt +
βθw
ζwEtπw
t+1 +ϕ
ζwπw
t−1
(27)
πt =(1− ω)(1− θ)(1− βθ)
ζXt +
βθ
ζEtπt+1 +
ω
ζπt−1 (28)
Yt = At + (1− α)Lt (29)
wrt = Xt + At − αLt (30)
∆wrt =πw
t − πt (31)
Xt =
[χ+ α
1− α+ σ
](Yt − Y ∗t
)−[χLt + σYt − wr
t
](32)
At = ρAt−1 + ut (33)
Here:
µt ≡ χLt + σYt − wrt
ζw ≡ θw + ϕ[1− θw(1− β)]
ζ ≡ θ + ω[1− θ(1− β)]
Upper case letters denote the aggregate of the respective lower case variables.(27) and (27) are the wage and price Philips curves. (29) is the log-linearizedproduction function. It has been pointed out by Yun (1996) that the full non-linearaggregate production function depends on a price dispersion term.
Yt =At
Dt
KαL1−αt with: Dt ≡
∫ 1
0
(Pt(z)
Pt
)−ε
dz and Dt = θDt−1 (34)
7The consumption Euler equation does not impose a constraint on the policymaker. It servesmerely to back out the path of the nominal instrument that supports the optimal allocation forgiven optimal paths of output and inflation.
9
Christiano, Eichenbaum, and Evans (2001) have shown that the price dispersionterm can be ignored for a loglinear analysis around a steady state with zero pricedispersion. One can further show that this term evolves as a univariate AR(1)regardless of the fraction of backward looking price setters by log-linearizing theprice index and the price dispersion term. (30) is the firm’s labor demand function.(31) is an identity defining the change the of the real wage. (31) links marginal costto the output gap and the ”wage gap”. That wage gap is the difference betweenthe average marginal rate of substitution between consumption and labor on thehand and the real wage on the other, see proposition 1 in the appendix. Finally,the last equation is the exogenous stochastic process is total factor productivity.
The welfare measure of the central bank is expected discounted lifetime utilityof a randomly drawn household. As is common in the literature, we neglect thearbitrarily small utility flow from real money balances.
E0
∞∑j=0
βjWt+j ≡ E0
∞∑j=0
βj
{U(Ct+j) +
∫ 1
0
V (Nt+j(h))dh
}. (35)
where the unconditional expectationE averages across all possible histories ofaggregate shocks. LetW∗
t denote period utility under perfectly flexible wagesand prices. Following proposition 5 in the appendix, the consumption equivalentwelfare measureL ≡ −
∑∞t=0 β
t (Wt −W∗t ) /
(UCC
)can be approximated up to
second order by the following weighted sum of second moments.
L = E0
∞∑t=0
βt
[λ0π
2t + λ1
(Yt − Y ∗t
)2
+2 +λ2 (∆πt)2 + λ3πw
2
t + λ4
(∆πw
t
)2](36)
Since this loss function is free of first moments, it can be accurately evaluated byconsidering a linear approximation to the models equilibrium conditions. Here,the weights are given by
λ0 = 0.5εθ
(1− θ)(1− θβ)(37)
λ1 = 0.5
(χ+ α
1− α+ σ
)(38)
λ2 =ω
(1− ω)θλ0 (39)
λ3 = 0.5κ2(1− α)(κ−1 + χ)θw
(1− θw)(1− θwβ)(40)
λ4 =ϕ
(1− ϕ)θw
λ3 (41)
10
For the case of immobile capital,λ0 = 12
[1
1−α− ε−1
ε
]ε2 θ
(1−θ)(1−θβ)andλ2 adjust
accordingly. Any number for the loss function has no economic interpretation.However, the difference between two numbers when comparing alternative policyrules is an approximation of the one off increase in consumption (as a fraction ofsteady state consumption) necessary to make an average agent equally well offunder both policies.8
4 Calibration
We use the baseline calibration from the sticky price model of Pappa (2004) andassume symmetric price and wage setting parameters. In particular, the markupis 14 % in both goods and labor market resulting inκ = ε = 7.88. Furthermoreassume that prices and wages are fixed on average 4 quarters, such thatθ = θw =34. Set the fraction of backward looking agents in both price and wage setting
to 0. Furthermore assume the coefficient of relative risk aversionσ = 2 andassume a Frisch (constant marginal utility of wealth) elasticity of labor supply of13, implyingχ = 3. The exogenous process for technology follows an AR(1) with
autoregressive parameter equal to0.906. The innovation has standard deviationequal to0.00852. Finally the time preference rate is matched to yield an annualreal interest rate of1.03, i.e. β = 1.03−0.25 These parameters give rise to thefollowing weights in the loss function for the case of perfectly mobile capital:
λ0 = 23.30, λ1 = 6.21, λ3 = 200.91 (42)
Note that the weight on wage inflation is almost an order of magnitude larger thanon price inflation for our benchmark calibration despite the fact that the averageduration of wage contracts is the same as for price contracts. This is a result of alow wage elasticity. With stick wages, labor supply is demand determined. Theinverse of the labor supply elasticity signals how much compensation in terms ofreal wage the household requires for supplying an extra unit of labor. With stickywages households are induced to vary their labor supply without any such com-pensation taking place.9 Therefore, it is clear that the inverse of the labor supply
8Note that by dividing by the marginal utility of consumption (which has dimension utils perunit of consumption) we are expressing welfare in terms of units of consumption. Further divid-ing by steady state consumption we express the measure as percentage compensation necessaryto achieve the same level of welfare as under flexible prices and wages. It should be noted how-ever, that the derivation of this consumption equivalent welfare measure involved dropping termswhich are independent of policy. Therefore any single consumption equivalent number has noeconomic meaning. The difference between any two numbers is meaningful, as the omitted termsindependent of policy would drop out anyways once we form the difference.
9The assumption of complete consumption insurance implies that household are free of anyincome risk stemming from wage stickiness. Dropping this assumption, would further increase
11
elasticity is closely related to the welfare cost of nominal wage stickiness. For in-stance, settingχ = 1 brings the weight on wage inflation relative to price inflationdown to2.8 for our benchmark calibration. Another important parameter deter-mining the relative weight is the wage elasticity of labor demandκ. The higherthis parameter, the more substitutable are different varieties of labor in production.Differences in relative quantities of labor demanded by the labor aggregator are afunction of differences in relative wages posted and that function is increasing inthe substitutability (κ) of labor varieties in the aggregator. For instance, reducingthe markup in both labor and goods market to10% (κ = ε = 11) increases theweight on wage inflation stabilization to15.75
5 Comparison of monetary policy rules
This section analyzes both simple monetary policy rules and fully optimal policy.The system of first-order conditions for the fully optimal policy problem can befound in the appendix on page 33. We restrict the analysis to monetary policyunder commitment, because the rules we consider are very simple and thereforeeasy to communicate. Furthermore, the analysis of discretion vs. commitmentwith backward looking elements in price setting has already been conducted inSteinsson (2003), we expect results to be similar in this setup.
The measure of welfare we consider is the expectation as of time zero10 ofexpected discounted lifetime utility as in (36). In the following table, for anyvariablex V[x] ≡ E0
∑∞j=0 β
jx2t+j
We start the analysis by considering the fully optimal rule and three simplerules for a range of fractions of forward and backward looking wage and pricesetters. The simple rule we consider are complete output gap stabilizationV[Yt −Y ∗t ] = 0, complete wage inflation stabilizationV[πw
t] = 0 and complete priceinflation stabilizationV[πt] = 0.
5.1 Optimal simple rules
In this subsection, we consider a class of simple interest rate rules of the followingform
it = α0Gt + α1πt + α2πwt + α3it−1 (43)
the welfare costs of wage stickiness.10We assume the economy is in the steady at time zero.
12
We maximize welfare numerically subject to the condition that the equilibriumbe determinate11. We consider how welfare is affected by successively restrictingthe rules to respond to less variables. We eliminate the output gap from the rule,as authors such as Schmitt-Grohe and Uribe (2004) and Neiss and Nelson (2003)have noted that the theoretically correct measure of output gap (actual minus flex-ible price output) is difficult to obtain in practice and may be badly proxied bynon model based concepts such as detrended output. Finally, Schmitt-Grohe andUribe (2004) have shown that responding to an incorrect measures of the gap,such as deviation from steady state, can involve large welfare losses . Given thisrisk, it seems natural ask how rules perform that neglect the output gap altogether.We furthermore eliminate the lagged interest rate from the reaction function inorder to measure the gains from inertia. It has often been argued that reacting tothe lagged interest rate is a simple way to introduce history dependence into thepolicy rate. Such inertia in simple rules may mimick the history dependence thatis an important feature of the fully optimal plan under commitment with forwardlooking agents. Finally, we consider two rules that have often been proposed inthe literature: The first one the classic Taylor (1993) rule:
it = 0.5Gt + 1.5πt (44)
Finally, Amato and Laubach (2003a) found that a first difference version ofthis rule performed well in a model with rule of thumb price setters and consumersand it seems natural to consider it here as well. In that version of the rule, the rateof change of the nominal interest rate responds the output gap and price inflationwith coefficients as suggested in Taylor (1993).
∆it = 0.5Gt + 1.5πt (45)
Before discussing the performance of these simple rules, we undertake an evensimpler analysis and compute the welfare costs of rules that fully stabilized one ofthree variables: price level, wage level and output gap. The detailed results are de-ferred for the appendix on page 35. Note that in the absence of wage rigidities, fullstabilization of the price level or equivalently the output gap is the optimal rule.12
11Since equilibrium selection under indeterminacy is controversial, we require that the optimalrules implies determinacy. The algorithm assigns an arbitrarily large loss to rules that renderthe equilibrium indeterminate. Therefore the loss function is discontinous at the boundary of thedeterminacy region and standard MATLAB (Version 6.0) minimization routines are not applicable.We use the ”cliff-robust” minimization routinecsminwel.m, that can deal with such setups. It isprovided by C. Sims athttp://eco-072399b.princeton.edu/yftp/optimize
12This follows from the absence of any time varying inefficiencies such that price inflation andthe output gap are linearly related, see equation (17). Therefore, achieving zero variance for priceinflation implies zero variance for the output gap and welfare under sticky prices is equal to welfareunder flexible prices.
13
It turns out that the rules that are optimal when only prices are sticky, involve largelosses when wages are sticky additionally. The loss from full price level stabilityranges fromXXX to XXX for varying degrees of backward looking agents inwage and price setting. ....
In the following table, coefficients with an asteriks as superscripts are imposedas a restriction, all other coefficients are optimized
6 Specific factor markets
So far we have assumed that capital is freely mobile across sectors and can bereallocated as to equalize the shadow value of capital across firms. It has beenargued by Danthine and Donaldson (2002), Woodford (2003, p.166) and othersthat capital cannot be instantaneously be relocated across firms. In particular, itappears to be highly unreasonable that it is too costly to post a new price tag,but that it is costless to unbolt machinery and ship it between firms. Further-more, Eichenbaum and Fischer (2004) show that departing from the assumptionof perfect capital mobility is necessary to reconcile the Calvo (1983) model withthe data. Sveen and Weinke (2004) further discuss the implications of modelingcapital for the equilibrium dynamics in sticky prices models
For the purpose of business cycle analysis, capital might better be modeled asbeing firm specific. In this subsection, we solve the firms price setting problemwhen capital is fixed at the firm level. The problem of the firm is now to choosePt(z) subject to the demand curve and the production function to maximize
maxP ∗t (z)
Et
∞∑i=0
(θβ)iΛt,t+i
{(1 + τp)
[P ∗t (z)
Pt+i
]1−ε
Yt+i − Zt+iK(z)− wrt+iLt+i(z)
}.
(46)
Noting that ∂Lt+j(z)
∂Pt(z)=
∂Lt+j(z)
∂Yt+j(z)
∂Yt+j(z)
∂Pt(z)= − ε
1−α
Lt+j(z)
Pt(z), the first order condition
for this problem is
Et
∞∑i=0
(θβ)iΛt,t+i
{(1 + τp)(1− ε)
[P ∗tPt+i
]1−ε
Yt+i +ε
1− αwr
t+iLt+i(z)
}(47)
With firm specific capital, proposition 4 in the appendix shows that the PhilipsCurve is given by
πt =(1− ω)(1− θ)(1− βθ)
ζ
(1− α)
(1− α+ αε)Xa
t +βθ
ζEtπt+1 +
ω
ζπt−1 (48)
14
Table 1: Optimal and simple rules - mobile capital
(ω, ϕ) rule α0 α1 α2 α3 L
(0, 0)
optimal - - - - 3.441
rule 1 9.48 6.49 8.57 3.05 3.444
rule 2 0∗ 4.21 20.96 0.81 3.444
rule 3 0∗ 141.85 951.19 0∗ 3.492
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 27.826
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 4.308
(12, 0)
optimal - - - - 3.678
rule 1 5814.47 2937.22 -80.66 1392.04 3.680
rule 2 0∗ 9.56 37.45 -0.73 3.767
rule 3 0∗ 5.85 21.55 0∗ 3.779
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 28.37
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 4.557
(0, 12)
optimal - - - - 3.557
rule 1 0.11 1.66 7.75 0.98 3.561
rule 2 0∗ 1.60 7.70 0.96 3.561
rule 3 0∗ 4.56 28.16 0∗ 4.021
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 32.488
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 4.657
(12, 1
2)
optimal - - - - 3.826
rule 1 7.70 13.45 51.62 2.25 3.831
rule 2 0∗ 5.17 26.93 0.87 3.835
rule 3 0∗ 11.26 61.00 0∗ 3.893
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 32.850
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 5.02
15
Here, Xat is average marginal cost. Since capital is fixed at the firm level, the
capital-labor ratio differs across firms and so does marginal cost.When we assume that capital is fixed at the level of an individual firm, the
weight on price inflation in the loss function rises roughly by a factor 15 to364.23,while the weights on the variability of the output gap and wage inflation remainthe same. Price dispersion becomes much more costly now, since costs of outputdispersion across producers rise. Price dispersion implies that the bundler de-mands different varieties in relative quantities that are socially inefficient. Withfirm specific capital, we have an additional inefficiency. Now each firm is produc-ing the wrong quantities with the ”wrong” mix of factor inputs. For both commonand specific capital, a given dispersion of relative prices leads to a the same dis-persion of relative quantities. However, firm specific capital implies that a givendispersion in relative quantities results in a much bigger dispersion of labor acrossfirms. That follows from firms inability to re-allocate capital to produce with theefficient capital labor ratio. Capital is fixed at the firm level, the firm can onlyadjust labor to vary production. Since labor has decreasing marginal product inproduction at the level of the individual firm, the dispersion of labor across firms iswelfare reducing.13 Therefore, the weight attached to price inflation rises stronglywith firm specific capital.
A greater weight on the price inflation variability does not imply that priceinflation targeting is more desirable with firm specific capital than with mobilecapital. The reason is that the structural equations change, too. In particular,the slope of the price Philips curve falls by factor 15 from0.0852 to 0.0055. Agiven disturbance to marginal cost results in much less price inflation with firmspecific capital. Therefore, it may very well be the case that strong wage inflationtargeting remains a desirable policy despite the fact that price inflation receives amuch higher weight in the loss function with firm specific capital.
We now consider the performance of simple rules for the case of immobilecapital. As noted earlier, immobile capital strongly raises the weight on priceinflation variability in the loss function. At the same time it, the price Philipscurve implies that a given disturbance to marginal cost has much less of an impacton price inflation. It is therefore a priori unclear how the results from the analysisof freely mobile capital will change.
13If production were linear in labor, the weights attached to price inflation variability would bethe same across mobile and firm specific capital. Dispersion of labor across firms would still bewelfare reducing, but only because it is identical to the dispersion of output across firms. Sinceeach variety has decreasing marginal product in the bundler, dispersion of output is again welfarereducing.
16
Table 2: Optimal and simple rules - immobile capital
(ω, ϕ) rule α0 α1 α2 α3 L
(0, 0)
optimal - - - - 2.636
rule 1 11.95 4.82 5.55 1.97 2.658
rule 2 0∗ 231,697.27 299,570.44 4,816.31 2.661
rule 3 0∗ 54,935.89 69,025.37 0∗ 2.682
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 21.746
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 2.855
(12, 0)
optimal - - - - 3.177
rule 1 124.28 318.82 344.55 21.70 3.219
rule 2 0∗ 32.23 25.64 -0.84 3.446
rule 3 0∗ 18.41 14.13 0∗ 3.465
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 26.047
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 3.401
(0, 12)
optimal - - - - 2.749
rule 1 143.39 26.77 43.87 2.86 2.754
rule 2 0∗ 6.40 8.50 1.19 2.783
rule 3 0∗ 19.65 23.09 0∗ 3.14
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 24.869
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 3.115
(12, 1
2)
optimal - - - - 3.349
rule 1 29.09 11.49 4.02 3.10 3.365
rule 2 0∗ 453.98 543.12 2.80 3.379
rule 3 0∗ 351.36 409.14 0∗ 3.381
Taylor 0.5∗ 1.5∗ 0∗ 0∗ 29.410
Taylor FD 0.5∗ 1.5∗ 0∗ 1∗ 3.694
17
7 Summary and Conclusion
This paper has evaluated the welfare effects of monetary policy rules in a simplegeneral equilibrium model with sticky wages and prices. It has analyzed fullyoptimal policy and simple monetary policy rules in variants of the baseline modelof Erceg, Henderson, and Levin (2000). We chose our departures from the seminalpaper at modeling points with very little consensus among macro-economists.
First, we depart from the assumption of full rationality and purely forwardlooking behavior by allowing a fraction of wage and price setters to be backwardlooking. This is in the spirit of rule of thumb consumers as in Campbell andMankiw (1989) test of the life cycle hypothesis and was first suggested for pricesetters by Galı and Gertler (1999). Second, following the criticism in Danthineand Donaldson (2002), we depart from the assumption of a frictionless rentalmarket for capital that instantaneously and costlessly allows to reallocate capitalacross firms . Instead we model capital as fixed at business cycle frequency. Fi-nally, we scrutinize Calvo (1983) price and wage contracts. It has been pointedout by Kiley (2002) and Ascari (2004) that Calvo (1983) contracts imply muchmore price dispersion than comparable schemes with finite horizon. We allowfor a general contract scheme as suggested in Wolman (1999) that encompassesTaylor (1980) contracts as a special case and can pick up some salient features ofstate dependent pricing of Dotsey, King, and Wolman (1999).
Introducing backward looking wage and price setters increases the cost of anygiven monetary policy rules, but only moderately. When some contracts are de-termined by backward looking agents, inflation is more persistent. A shock toinflation today affects inflation in future periods, because some agents are back-ward looking. While inflation is more persistent it is also less variable.
18
Appendix
Proposition 1 (marginal cost). Up to first order, marginal costs is related to thegaps driving the wage and price Philips curves in the following way
Xt =
[χ+ α
1− α+ σ
](Yt − Y ∗t
)−[χLt + σYt − wr
t
](49)
Proof of proposition 1: Log-linearize the first order condition for labor demand
wrt = −αLt + At + Xt (50)
Define µt ≡ χLt + σCt − wrt. Subtracting the marginal rate of substitution,
χLt + σYt, from both sides of the above expression, one obtains
−µt = [−χ− α] Lt − σYt + At + Xt (51)
Log-linearizing the production function, we can replaceLt with (1−α)−1(Yt−At)and arrive at
−µt =
[−χ− α
1− α− σ
]Yt +
[1 +
α+ χ
1− α
]At + Xt (52)
−µt = −[χ+ α
1− α+ σ
]Yt +
[χ+ α
1− α+ σ
] [1 + χ
χ+ α+ σ(1− α)
]At + Xt (53)
Recalling the definition of output under flexible prices (16), one can write
Xt =
[χ+ α
1− α+ σ
](Yt − Y ∗t
)− µt (54)
Proposition 2 (hybrid wage Philips curve). Under Calvo wage setting with ameasure of backward looking firms, the hybrid new wage Keynesian Philips curveis
πwt =
(1− ϕ)(1− θw)(1− βθw)
(1 + κχ)ζwµt +
βθw
ζwEtπw
t+1 +ϕ
ζwπw
t−1
Hereζw ≡ θw + ϕ[1− θw(1− β)].
Proof of proposition 2: We assume the wage subsidy is set to exactly offsetimpact of the monopolistic competition in the steady state1 + τ = κ
κ−1. We can
19
rewrite (6) using the demand functionNt(h) =(
Wt(h)Wt
)−κ
Lt andVN(Nt(h))t =
−Nt(h)χ as well asUC(Ct(h)) = C−σ
t to express it in terms of the optimal nomi-nal wageW ∗
t and aggregate variables only:
Et
∞∑j=0
(θwβ)j
(W ∗
t
Wt+j
)−κ
Lt+jC−σt+j
W ∗t
Pt+j
−
(W ∗
t
Wt+j
)−κχ
Lχt+j
C−σt+j
= 0 (55)
We can solve for the nominal wageW ∗t
W ∗t =
( ∑∞j=0(θwβ)jW
κ(1+χ)t+j L1+χ
t+j∑∞j=0(θwβ)jW κ
t+jLt+jC−σt+jP
−1t+j
) 11+κχ
(56)
Definew∗t ≡W ∗
t
Wt, wr
t ≡ Wt
Ptandπw
t,t+j ≡Wt+j
Wt. We can write the above condition
as
w∗t =
∑∞j=0(θwβ)j
(πw
t,t+j
)κ(1+χ)L1+χ
t+j∑∞j=0(θwβ)j
(πw
t,t+j
)κLt+jC
−σt+j
wrt+j
πwt,t+j
11+κχ
(57)
Rewrite this expression defining the following two auxiliary variables.
Dt ≡∞∑
j=0
(θwβ)j(πw
t,t+j
)κ(1+χ)L1+χ
t+j (58)
Gt ≡∞∑
j=0
(θwβ)j(πw
t,t+j
)κLt+jC
−σt+j
wrt+j
πwt,t+j
(59)
These infinite sums have a recursive representation. The behavior of optimizingfirms is fully described by the following three equations
w∗t =
(Dt
Gt
) 11+κχ
(60)
Dt =L1+χt + βθEt
(πw
t+1
)κ(1+χ)Dt+1 (61)
Gt =LtC−σt wr
t + βθEt
(πw
t+1
)κ−1Gt+1 (62)
Definew∗t =W ∗
t
Wtandwb
t =W b
t
Wt. Log-linearize the definition of the wage index
(25) and the rule of thumb (23), respectively:
wbt =
θw
(1− θw)ϕπw
t −(1− ϕ)
ϕw∗t (63)
wbt = (1− ϕ)w∗t−1 + ϕwb
t−1 − πwt + πw
t−1 (64)
20
Use the first equation to eliminatewbt from the second and solve forw∗t to obtain
w∗t =(1− θw)ϕ+ θw
(1− ϕ)(1− θw)πw
t −ϕ
(1− ϕ)(1− θw)πw
t−1 (65)
Log-Linearizing the auxiliary equations defining recursively the condition for op-timal wage setting around a steady state with zero wage inflation yields
Dt = (1− βθw)(1 + χ)Lt + βθκ(1 + χ)πwt+1 + βθwDt+1 (66)
Gt = (1− βθw)[wr
t + Lt − σCt
]+ βθ(κ− 1)πw
t+1 + βθwGt+1 (67)
Substituting these two equations into the log-linearized first order condition forwage setting (60) yields
w∗t =(1− θwβ)
(1 + κχ)
[χLt + σCt − wr
t
]+ βθw
(πw
t+1 + w∗t+1
)(68)
Substituting outw∗t one arrives at
πwt =
(1− ϕ)(1− θw)(1− βθw)
(1 + κχ)ζwµt +
βθw
ζwEtπw
t+1 +ϕ
ζwπw
t−1 (69)
Here,µt ≡ χLt + σCt − wrt andζw ≡ θw + ϕ[1− θw(1− β)].
Proposition 3 (hybrid price Philips curve). Under Calvo price setting with ameasureω of backward looking firms, the hybrid new Keynesian price Philipscurve is
πt =(1− ω)(1− θ)(1− βθ)
ζXt +
βθ
ζEtπt+1 +
ω
ζπt−1
Hereζ ≡ θ + ω[1− θ(1− β)].
Proof of proposition 3: Focussing on a symmetric equilibrium and thereforedroppingz, (14) can be expressed as
P ∗t =Et
∑∞j=0(θβ)iΛt,t+iXt+iP
εt+iYt+i
Et
∑∞j=0(θβ)iΛt,t+iP
ε−1t+i Yt+i
(70)
It is again convenient to rewrite the Calvo price setting condition in terms of sta-tionary variables using these auxiliary equations
p∗t =Bt
Ft
, with: (71)
Bt ≡ Et
∞∑i=0
(θβ)iΛt+iXt+i
(Pt+i
Pt
)ε
Yt+i, (72)
Ft ≡ Et
∞∑i=0
(θβ)iΛt+i
(Pt+i
Pt
)ε−1
Yt+i. (73)
21
p∗t stands for the optimal nominal priceP ∗t divided by the current period priceindexPt. The infinite discounted sumsBt andFt have a recursive representation,where we have expressed the stochastic discount factorΛt,t+i by the ratio of themarginal utilities of consumption:
Bt = XtYt + βθEtC−σt+1C
σt π
εt+1Bt+1, (74)
Ft = Yt + βθEtC−σt+1C
σt π
ε−1t+1Ft+1. (75)
Definep∗t =P ∗tPt
andpbt =
P bt
Pt. Log-linearize the definition of the price index (20)
and the rule of thumb (18), respectively:
pbt =
θ
(1− θ)ωπt −
(1− ω)
ωp∗t (76)
pbt = (1− ω)p∗t−1 + ωpb
t−1 − πt + πt−1 (77)
Use the first equation to eliminatepbt from the second and solve forp∗t to obtain
p∗t =(1− θ)ω + θ
(1− ω)(1− θ)πt −
ω
(1− ω)(1− θ)πt−1 (78)
The log-linearized first order condition for price setting of forward looking firms(71), (74) and (75) around a steady state with zero price inflation yields
p∗t = (1− θβ) Xt + βθ(πt+1 + p∗t+1
)(79)
Substituting outp∗t one arrives at
πt =(1− ω)(1− θ)(1− βθ)
ζXt +
βθ
ζEtπt+1 +
ω
ζπt−1 (80)
Hereζ ≡ θ + ω[1− θ(1− β)].
Proposition 4 (Price Philips curve with specific factor markets).Under Calvoprice setting with a measureω of backward looking firms, the hybrid new Keyne-sian price Philips curve for the case of specific factor markets is
πt =(1− ω)(1− θ)(1− βθ)
ζXt +
βθ
ζEtπt+1 +
ω
ζπt−1
Hereζ ≡ θ + ω[1− θ(1− β)].
Proof of proposition 4: Using the production function to express hours in termsof output, using the demand function, recalling that the subsidy offsets the steady
22
state effects of monopolistic competition and definingΨt ≡ Y1
1−α
t+i A− 1
1−α
t+i K−α1−α
we can rewrite the first order condition for price setting as
Et
∞∑i=0
(θβ)iΛt,t+i
{[P ∗tPt+i
]1−ε
Yt+i +1
1− αwr
t+i
[P ∗tPt+i
]− ε1−α
Ψt+i
}(81)
Solving for the optimal nominal price yields
P ∗t =
Et
∑∞i=0(θβ)iΛt,t+i
11−α
wrt+iP
ε1−α
t+i Ψt+i
Et
∑∞i=0(θβ)iΛt,t+iPt+i
ε−1Yt+i
1−α1−α+εα
(82)
Dividing trough by the current price level, we can express this in terms of station-ary variables14
p∗t =
[Et
∑∞i=0(θβ)iΛt,t+i
11−α
wrt+i
(Πi
j=1πt+j
) ε1−α Ψt+i
Et
∑∞i=0(θβ)iΛt,t+i
(Πi
j=1πt+j
)ε−1Yt+i
] 1−α1−α+εα
(83)
Rewriting these expressions recursively, by use of auxiliary variables
p∗t =
(Bt
Ft
) 1−α1−α+εα
, with: (84)
Bt ≡ Et
∞∑i=0
(θβ)iΛt,t+i1
1− αwr
t+i
(Πi
j=1πt+j
) ε1−α Ψt+i (85)
Ft ≡ Et
∞∑i=0
(θβ)iΛt,t+i
(Πi
j=1πt+j
)ε−1Yt+i. (86)
The infinite discounted sumsBt andFt have the following recursive representa-tion
Bt =1
1− αwr
t Ψt + βθEtC−σt+1C
σt π
ε1−α
t+1 Bt+1, (87)
Ft = Yt + βθEtC−σt+1C
σt π
ε−1t+1Ft+1. (88)
Log-linearizing (84), (87), and (88) yields the following equation
p∗t =
(1− α
1− α+ αε
)(1− θβ)
[wr
t +1
1− α
(αYt − At
)]+ βθ
(πt+1 + p∗t+1
)(89)
14Π0j=1πt+j ≡ 1.
23
Note that for any firmz loglinear marginal cost (i.e the ratio of the real wage
to the marginal product of labor) iswrt + 1
1−α
(αY (z)t − At
). Since firms post
difference prices, they sell different quantities, and marginal cost differs acrossfirms. Up to log-linearization, the output aggregator equals the average over itsindividual components, i.e
∫ 1
0Y (z)tdz = Yt +O(||ξ||2). Therefore,Xa
t ≡ wrt +
11−α
(αYt − At
)is a first order approximation of average marginal cost.
Substituting outp∗t and using steps similar as for specific factor markets, onearrives at
πt =(1− ω)(1− θ)(1− βθ)
ζ
(1− α)
(1− α+ αε)Xa
t +βθ
ζEtπt+1 +
ω
ζπt−1 (90)
Proposition 5 (loss function).Define average utility across households
Wt = U(Ct+j) +
∫ 1
0
V (Nt+j(h))dh (91)
Let W∗t denote average utility under flexible prices and wages. With Calvo wage
and price setting, we can approximate the loss functionL ≡ −∑∞
t=0 βt (Wt −W∗
t ) /UCCup to second order and neglecting terms independent of policy by the term
L = E0
∞∑t=0
βt
[λ0π
2t + λ1
(Yt − Y ∗t
)2
+2 +λ2 (∆πt)2 + λ3πw
2
t + λ4
(∆πw
t
)2](92)
The weights are given by
λ0 =1
2ε
θ
(1− θ)(1− θβ)
λ1 =1
2
(χ+ α
1− α+ σ
)λ2 =
ω
(1− ω)θλ0
λ3 =1
2(1− α)(κ−1 + χ)κ2 θw
(1− θw)(1− θwβ)
λ4 =ϕ
(1− ϕ)θw
λ3
Proof of proposition 5 (following Erceg, Henderson, and Levin (2000)): Somerelations are used repeatedly throughout this text. For a generic variableXt, let
24
Xt ≡ Xt − X denote arithmetic deviation from the steady stateX, let Xt ≡logXt − logX denote logarithmic deviation. Up to second order, the relation be-tween the two is
Xt ≈ X
(Xt +
1
2X2
t
).
For the often used Dixit-Stiglitz type agrregators of the form
Xt =
[∫ 1
0
Xt(j)φdj
] 1φ
the logarithmic approximation is
Xt ≈EjXt(j) +1
2φVARjXt(j). (93)
Here, the cross-sectional mean is denoted byEj and the cross-sectional varianceis denoted byVARj. Finally, let ||ξ|| denote an upper bound to the exogenousdisturbances. The goal of the following paragraphs is to approximate all expres-sions involving integrals across households indexed byh or across firms indexedby z, in terms of aggregate variables. Approximations are second order Taylorexpansions, i.e. terms of order higher than two are omitted.15
Equipped with these simple tools, (35) can be approximated in the followingway. The second order approximation to the utility of consumption is
U(Ct) = UCCCt +1
2
(UCC + UCCC
2)C2
t +O(||ξ||3). (94)
Here,O(||ξ||3) denotes a residual that is third order or higher in the bound on theexogenous disturbance. Taking unconditional expectation16
EU(Ct) = UCC
[ECt +
1
2(1 + ω1)VAR Ct
]+O(||ξ||3). (95)
Hereω1 = UCC CUC
is the elasticity of marginal utility of consumption evaluated atthe steady state. The second order approximation to the utility of labor is
V (Nt(h)) = VNNNt(h) +1
2
(VNN + VNNN
2)N2
t (h) +O(||ξ||3)
15Omitting all terms of order higher than two implies that when substitutions into squares ofvariables are undertaken, only the first order terms of the Taylor expansion of these variables aresubstituted.
16We are using the fact that up to first orderCt has mean zero, so that up to second order thecentered and uncentered second moments are equal, i.e.EC2
t = (EC)2 + VARCt = VARCt +O(||ξ||3).
25
Integrating this expression overh yields∫ 1
0V (Nt(h))dh
VNN= EhNt(h) +
1
2(1 + ω2)
([EhNt(h)
]2+ VARhNt(h)
)+O(||ξ||3)
The termω2 ≡ VNN NVN
is the elasticity of marginal utility of labor evaluated at the
steady state. From total labor supplyLt ≡[[Nt(h)]
κ−1κ dh
] κκ−1
using the results
stated in (93)
Lt = EhNt(h) +κ− 1
2κVARhNt(h) +O(||ξ||3) (96)
Solve (96) to eliminateEhNt(h). This yields the following approximation to theutility of labor∫ 1
0
V (Nt(h))dh =VNN
[Lt +
1
2(1 + ω2)L
2t
](97)
+1
2VNN
[(κ−1 + ω2)VARhNt(h)
]+O(||ξ||3) (98)
We need to eliminateLt in order to arrive at an output gap term, consider thesecond order approximation to total demand for laborLt =
∫ 1
0Lt(z)dz
Lt ≡ logEzLt(z)− logL = EzLt(z) +1
2VARzLt(z) +O(||ξ||3) (99)
Since all firms face the same relative price of capital and the labor bundler, theyhave the same capital-labor ratio. This fact, together with a fixed aggregate capitalstock gives rise to the following exact loglinear relation derived from the produc-tion function.
EzLt(z) = EzYt(z)− At + αLt (100)
VARzLt(z) =VARzYt(z) (101)
From the definition of the output bundler, we have
EzYt(z) = Yt −1
2
ε− 1
εVARzYz(z) +O(||ξ||3) (102)
Substituting (100) into yields (99)
Lt = EzYt(z)− At + αLt +1
2VARzYt(z) +O(||ξ||3) (103)
26
Using (102) to eliminateEzYt(z) and rearranging yields
Lt =1
1− α
(Yt − At +
1
2εVARzYt(z)
)+O(||ξ||3) (104)
Substituting forLt andL2t into (97) yields
2
VNN
∫ 1
0
V (Nt(h))dh =2
1− α
(Yt − At
)+
1 + ω2
(1− α)2
(Yt − At
)2
+1
(1− α)εVARzYt(z) (105)
+ (κ−1 + ω2)VARhNt(h) +O(||ξ||3)
Combining the approximations of the sub-utility function for consumption and forlabor, yields
W =UCC
[ECt +
1
2(1 + ω1) EC2
t
]+VNN
1− αE(Yt − At
)+VNN
2
1 + ω2
(1− α)2E(Yt − At
)2
+VNN
2
1
(1− α)εVARzYt(z)
+VNN
2(κ−1 + ω2)VARhNt(h) +O(||ξ||3)
The model features no aggregate capital accumulation, henceCt = Yt. Further-more, since the labor-leisure decision is not distorted in the steady state, we havethat(1−α)UC Y = −VNN .17 Together these two conditions ensure that all linearterms cancel, except terms which are independent of policy and therefore do notaffect the optimal policy. These terms are denotedt.i.p.
W0.5UCC
=
((1 + ω1)−
1 + ω2
(1− α)
)E Y 2
t + 21 + ω2
(1− α)E YtAt −
1
2εVARzYt(z)
− 2(1− α)(κ−1 + ω2)VARhNt(h) + t.i.p.+O(||ξ||3)
As such the above model is free of first order terms and can be accurately eval-uated using a linear approximation to the models equilibrium conditions. It hasbecome customary in the literature to rewrite the expressions in terms of the out-put gapYt − Y ∗t . This aids in our economic interpretation of the policy problem.Subtracting from this equationW∗, the second order approximation to the utility
17This follows from the first order condition for labor supplyUCwp = −VN andw
p = (1−α) YN
27
function evaluated in a model with completely flexible prices and wages and anefficient level of outputY ∗t , one arrives at18
W−W∗
UCC=
1
2
((1 + ω1)−
1 + ω2
(1− α)
)E[Y 2
t − (Y ∗t )2]
+1 + ω2
(1− α)EAt
(Yt − Y ∗t
)− 1
2εVARzYt(z)
− 1
2(1− α)(κ−1 + ω2)VARhNt(h) + t.i.p.+O(||ξ||3)
To rewrite the expression in terms of an output gap, we proceed in the followingway. Using the definition of the natural level of output, and defining the termΛ ≡ ω2 + α− (1− α)ω1 we have the following two relations
1
2
((1 + ω1)−
1 + ω2
(1− α)
)E[Y 2
t − (Y ∗t )2]
= − Λ
2(1− α)
[Y 2
t − (Y ∗)2],
(106)1 + ω2
(1− α)At
(Yt − Y ∗t
)=
Λ
1− αY ∗t (Yt − Y ∗t ). (107)
Finally note that
−1
2
(Yt − Y ∗
)2
= −1
2
[Y 2
t − (Y ∗)2]
+ Y ∗t (Yt − Y ∗t ) (108)
Therefore, we can rewrite the welfare criterion compactly as
W−W∗ =λ1VAR(Yt − Y ∗t
)+ λ2VARhNt(h) + λ3VARzYt(z) + t.i.p+O(||ξ||3)
(109)
Here, the weights are given by:
λ1 = −1
2
UCCΛ
(1− α), λ2 = −1
2(1− α)(κ−1 + ω2)UCC, λ3 = −1
2
UCC
ε. (110)
Define a loss function asL ≡ −∑∞
j=0 βj(Wt+j −W∗
t+j
)/(UCC). Further, de-
fine ∆Nt ≡ VARh logNt(h) and similarly∆Y
t ≡ VARz log Yt(z). Note thatit follows from proposition 8 that the infinite discounted sum of cross-sectionaldispersion of output across producers can be rewritten as
∞∑j=0
βj∆Yt+j =
∞∑j=0
βj ε2
(1− βθ)(1− θ)
[θ (log πt)
2 +ω
(1− ω)(∆ log πt)
2
]+O(||ξ||3) + t.i.p. (111)
18Since policy has no effect on welfare with completely flexible prices, this is equivalent toadding a constant in any maximization problem: It will change the value of the objective function,but not the value of the maximand.
28
Similarly, it follows from proposition 7 that∞∑
j=0
βj∆Nt+j =
∞∑j=0
βj κ2
(1− βθw)(1− θw)
[θw (log πw
t )2 +ϕ
(1− ϕ)(∆ log πw
t )2
]+O(||ξ||3) + t.i.p. (112)
Using these expressions, we arrive at (92).
Proposition 6 (loss function with immobile capital). When capital is immobileat the firm level, we can approximate the loss functionL ≡ −
∑∞t=0 β
t (Wt −W∗t ) /UCC
by a loss function of the same form as with freely mobile capital. The weights inthe loss function attached to the variance of price inflation and the variance of therate of change of price inflation now change to
λ0 =1
2
[1
1− α− ε− 1
ε
]ε2
θ
(1− θ)(1− θβ)
λ2 =ω
(1− ω)θλ0
Proof of proposition 6: Here we sketch on the part of the derivation that is dif-ferent from the case with mobile capital. The following loglinear relation derivedfrom the production function holds exactly
EzYt(z) = At + (1− α)EzLt(z) (113)
It follows thatVARzLt(z) = 1(1−α)2
VARzYt(z). Solving forL(z)t, substitutinginto (99) and making use of (102) yields
Lt =1
1− α
[Yt − At
]+
1
2
[1
(1− α)2− ε− 1
ε(1− α)
]VARzYt(z) (114)
Substituting this into (97) yields
2
VNN
∫ 1
0
V (Nt(h))dh =2
1− α
(Yt − At
)+
1 + ω2
(1− α)2
(Yt − At
)2
+1
1− α
[1
1− α− ε− 1
ε
]VARzYt(z) (115)
+ (κ−1 + ω2)VARhNt(h) +O(||ξ||3)
Following similar steps as before one can show that the weight on dispersion ofoutput across producers is now given byλ3 = −1
2UCC
[1
1−α− ε−1
ε
]. Following
the same steps as for mobile capital we arrive at the new weights.
29
Proposition 7 (wage dispersion with backward looking wage setters).With ameasure of backward looking wage setters, the cross sectional dispersion of labor∆N
t ≡ VARh logNt(h) is related to wage inflationlog πwt ≡ logWt − logWt−1
in the following way
∆Nt = θw∆N
t−1 + κ2
[θw
(1− θw)(log πw
t )2 +ϕ
(1− θw)(1− ϕ)(∆ log πw
t )2
]+O(||ξ||3)
(116)
Proof of proposition 7: From the demand function for laborNt(h) =(
Wt(h)Wt
)−κ
Lt
we have that
VARh logNt(h) =κ2VARh logWt(h). (117)
Proceed further in the following steps. DefineWt ≡ Eh logWt(h). Note that
Wt − Wt−1 = Eh
[logWt(h)− Wt−1
](118)
= θwEh
[logWt−1(h)− Wt−1
](119)
+ (1− θw)[(
1− ϕ)(logW ∗t (h)− Wt−1
)+ ϕ
(logW b
t − Wt−1
)]= (1− θ)
[(1− ϕ)(logW ∗
t (h)− Wt−1) + ϕ(logW b
t − Wt−1
)](120)
Note that, the difference betweenWt andlogWt is second order and therefore upto first order, the left hand side of (121) is the log of wage inflation.
log πwt = (1− θw)
[(1− ϕ)(logW ∗
t (h)− Wt−1) + ϕ(logW b
t − Wt−1
)]+O(||ξ||2)
(121)
Note further that taking logs of the definition of the rule of thumb, recalling thatlog πw
t−1 − Wt−1 = −Wt−2 +O(||ξ||2) and using (121) we have immediately
logW bt (h)− Wt−1 =
1
(1− θw)log πw
t−1 +O(||ξ||2) (122)
Similarly take logs of the rule of thumb, solve forlogW ∗t−1 and subtractWt−2 on
both sides to arrive at
logW ∗t−1(h)− Wt−2 =
1
(1− ϕ)
{[logW b
t − log πwt−1 − Wt−2
]+ ϕ
(logW b
t−1 − Wt−2
)}(123)
30
Forwarding this equation one period recalling that the difference betweenlogWt
andWt is second order and making use of (122) twice, we arrive at
logW ∗t (h)− Wt−1 =
1
(1− θw)(1− ϕ)
[log πw
t − ϕ log πwt−1
]+O(||ξ||2) (124)
Next, consider the measure of wage dispersion∆Wt ≡ VARh logWt(h). Sine
Wt−1 is independent ofh we can write
∆Wt =VARh
[logWt(h)− Wt−1
](125)
= Eh
{[logWt(h)− Wt−1
]2}− (Eh logWt(h)− Wt−1)2 (126)
= θwEh
{[logWt−1(h)− Wt−1
]2}− (Wt − Wt−1)2 (127)
+ (1− θw)[(1− ϕ)(logW ∗
t (h)− Wt−1)2 + ϕ(logW b
t (h)− Wt−1)2]
= θw∆Wt−1 + (1− θw)(1− ϕ)(logW ∗
t (h)− Wt−1)2
+ (1− θw)ϕ(logW bt (h)− Wt−1)
2 − (Wt − Wt−1)2 (128)
Substituting (122) and (124) and simplifying, one arrives at
∆Wt = θw∆W
t−1 +θw
(1− θw)(log πw
t )2 +ϕ
(1− θw)(1− ϕ)(∆ log πw
t )2 +O(||ξ||3)
(129)
Finally solving for∆Nt by making use of (117) we arrive at (121).
Proposition 8 (price dispersion with backward looking price setters).With ameasure of backward looking price setters, the cross sectional dispersion of output∆Y
t ≡ VARz log Yt(z) is related to price inflationlog πt ≡ logPt − logPt−1 inthe following way
∆Yt = θ∆Y
t−1 + ε2[
θ
(1− θ)(log πt)
2 +ω
(1− θ)(1− ω)(∆ log πt)
2
]+O(||ξ||3)
(130)
Proof of proposition 8: The proof follows similar steps as for wage dispersionand is omitted.
Proposition 9 (output and labor dispersion with Wolman (1999) pricing).Un-der the Wolman (1999) pricing scheme with a maximum ofJ cohorts of firmscharging identical prices, whose fraction of the overall price index are denoted by
31
ωj, we have the following approximation of dispersion of output across producersand of labor across households
VARz log Y (z)t = ε2Jp−1∑j=0
ωpj (logPt,j − logPt)
2 +O(||ξ||3) (131)
VARh logN(h)t =κ2
Jw−1∑j=0
ωwj (logWt,j − logWt)
2 +O(||ξ||3) (132)
Proof of proposition 9: Note again from the demand function faced by an in-dividual producer thatVARz log Y (z)t = ε2VARz logP (z)t. Using that the dif-ference betweenlogPt andEz logPt(z) is second order and thatPt is independentof z, we have that
VARz log Y (z)t = ε2VARz (logP (z)t − logPt) (133)
= ε2Ez (logP (z)t − logPt)2 − ε2 (Ez logP (z)t − Pt)
2 (134)
= ε2Ez (logP (z)t − logPt)2 +O(||ξ||3) (135)
= ε2J∑
j=0
ωj (logPt,j − logPt)2 +O(||ξ||3) (136)
The last equality follows from the fact that all firms in cohortj charge the sameprice and that these cohorts have weights in the price index are also the the prob-abilities of any firmz charging pricePt,j. Analogous steps can be done to provethe second part of the proposition.
Corollary. It follows immediately, that forN period overlapping Taylor (1980)contracts, output dispersion can be approximated by
VARz log Y (z)t =ε2
N
J−1∑j=0
E (logPt,j − logPt)2 +O(||ξ||3) (137)
Proposition 10 (loss function with Wolman (1999) pricing).Under the Wol-man (1999) pricing scheme with a maximum ofJ cohorts of firms (assuming onlyforward looking wage and price setters) we have the following loss function
L =E0
∞∑t=0
βt
[λ0
Jp−1∑j=0
ωpj
(p∗t−j
)2+ λ1
(Yt − Y ∗t
)2
+ λ3
Jw−1∑j=0
ωwj
(w∗t−j
)2]
(138)
with: λ0 =1
2ε, λ1 =
1
2
(χ+ α
1− α+ σ
), λ3 =
1
2(1− α)(κ−1 + ω2)κ
2
32
Proof of proposition 10: This follows immediately by plugging in the resultsfrom proposition 9 on page 31 into (109).
The Lagrangian of the policy problem:
DefineGt ≡ Yt − Y ∗t substitute out theYt from the policy problem to reduce thedimension of the system
E0
∞∑t=0
βt[π2
t + λ1G2t + λ2 (∆πt)
2 + λ3πw2
t + λ4
(∆πw
t
)2]+
∞∑t=0
ψ1t β
t
[−πw
t +(1− ϕ)(1− θw)(1− βθw)
(1 + κχ)ζw
[−Xt +
(χ+ α
1− α+ σ
)Gt
]+βθw
ζwπw
t+1 +ϕ
ζwπw
t−1
]+
∞∑t=0
ψ2t β
t
[−πt +
(1− ω)(1− θ)(1− βθ)
ζXt +
βθ
ζπt+1 +
ω
ζπt−1
]+
∞∑t=0
ψ3t β
t[−wr
t + Xt + At − αLt
]+
∞∑t=0
ψ4t β
t[−∆wr
t + πwt − πt
]+
∞∑t=0
ψ5t β
t
[−Xt +
[χ+ α
1− α+ σ
]Gt −
[(χ+ σ(1− α)) Lt + σAt − wr
t
]]Here:
ζw ≡ θw + ϕ[1− θw(1− β)] andζ ≡ θ + ω[1− θ(1− β)].
33
Write the problem more compactly
E0
∞∑t=0
βt[π2
t + λ1G2t + λ2 (∆πt)
2 + λ3πw2
t + λ4
(∆πw
t
)2]+
∞∑t=0
ψ1t β
t[−πw
t + ξ1Xt + ξ2Gt + ξ3πwt+1 + ξ4πw
t−1
]+
∞∑t=0
ψ2t β
t[−πt + ξ5Xt + ξ6πt+1 + ξ7πt−1
]+
∞∑t=0
ψ3t β
t[−wr
t + Xt + At − αLt
]+
∞∑t=0
ψ4t β
t[−wr
t + wrt−1 + πw
t − πt
]+
∞∑t=0
ψ5t β
t[−Xt + ξ8Gt − ξ9Lt − σAt + wr
t
]Here, the coefficients are given
ξ1 = − (1− ϕ)(1− θw)(1− βθw)
(1 + κχ)ζw
ξ2 = − ξ1
(χ+ α
1− α+ σ
)ξ3 =
βθw
ζw
ξ4 =ϕ
ζw
ξ5 =(1− ω)(1− θ)(1− βθ)
ζ
ξ6 =βθ
ζ
ξ7 =ω
ζ
ξ8 =χ+ α
1− α+ σ
ξ9 =χ+ σ(1− α)
34
The first-order conditions for this problem are
Lt : 0 = −αψ3t − ξ9ψ
5t
Xt : 0 = ξ1ψ1t + ξ5ψ
2t + ψ3
t − ψ5t
wrt : 0 = −ψ3
t − ψ4t + βψ4
t+1
Gt : 0 = 2λ1Gt + ξ2ψ1t + ξ8ψ
5t
πt : 0 = 2πt + 2λ2 [(1 + β)πt − πt−1 − βπt+1]− ψ2t + βξ7ψ
2t+1 +
1
βξ6ψ
2t−1 − ψ4
t
πwt : 0 = 2λ3πw
t + 2λ4
[(1 + β)πw
t − πwt−1 − βπw
t+1
]− ψ1
t + βξ4ψ1t+1 +
1
βξ3ψ
1t−1 + ψ4
t
Some simple policy rules
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38
Table 3: Optimal and simple rules for varying degrees of backward looking agents
(ω, ϕ) rule V[Gt] V[πt] V[∆πt] V[πwt] V[∆πw
t] L
(0, 0)
optimal 0.351 0.431 0.271 0.026 0.015 3.252
P stabil 2.619 0 0 0.255 0.118 11.802
W stabil 1.786 1.055 0.746 0 0 5.990
G stabil 0 0.414 0.157 0.294 0.065 13.641
(14, 0)
optimal 0.340 0.381 0.147 0.028 0.016 3.400
P stabil 2.619 0 0 0.255 0.118 11.802
W stabil 1.806 0.957 0.419 0 0 6.412
G stabil 0 0.401 0.090 0.294 0.065 13.767
(12, 0)
optimal 0.316 0.346 0.077 0.030 0.016 3.457
P stabil 2.619 0 0 0.255 0.118 11.802
W stabil 1.844 0.879 0.220 0 0 6.568
G stabil 0 0.397 0.049 0.294 0.065 13.874
(0, 14)
optimal 0.388 0.454 0.283 0.022 0.008 3.370
P stabil 2.755 0 0 0.249 0.066 12.840
W stabil 1.786 1.055 0.746 0 0 5.990
G stabil 0 0.468 0.162 0.334 0.038 16.151
(14, 1
4)
optimal 0.375 0.404 0.154 0.024 0.008 3.528
P stabil 2.775 0 0 0.249 0.066 12.840
W stabil 1.806 0.957 0.419 0 0 6.412
G stabil 0 0.456 0.094 0.334 0.038 16.293
(12, 1
4)
optimal 0.346 0.368 0.081 0.026 0.009 3.593
P stabil 2.775 0 0 0.249 0.066 12.840
W stabil 1.844 0.879 0.220 0 0 6.568
G stabil 0 0.454 0.052 0.334 0.038 16.427
(0, 12)
optimal 0.463 0.475 0.287 0.019 0.004 3.447
P stabil 3.149 0 0 0.254 0.035 13.965
W stabil 1.786 1.055 0.746 0 0 5.990
G stabil 0 0.646 0.174 0.464 0.022 22.666
(14, 1
2)
optimal 0.450 0.425 0.157 0.021 0.004 3.613
P stabil 3.149 0 0 0.254 0.035 13.965
W stabil 1.806 0.957 0.419 0 0 6.412
G stabil 0 0.638 0.103 0.465 0.022 22.870
(12, 1
2)
optimal 0.413 0.391 0.083 0.022 0.004 3.694
P stabil 3.149 0 0 0.254 0.035 13.965
W stabil 1.844 0.879 0.220 0 0 6.568
G stabil 0 0.646 0.059 0.467 0.022 23.115
39